We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law (1/rα) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic bound states in high-energy physics. We show that these bound states have dramatic consequences for the non-equilibrium dynamics following a global quantum quench, such as suppressed spreading of quantum information and oscillations of order parameters. The masses of these bound states can be read out from the Fourier spectrum of these oscillating order parameters. We then use a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement. The masses of the bound states predicted by this model are in good quantitative agreement with exact diagonalization results. Moreover, we illustrate that these bound states lead to weak thermalization of local observables for initial states with energy near the bottom of the many-body energy spectrum. Our work is readily applicable to current trapped-ion experiments.

Experimental advances have allowed for the exploration of nearly isolated quantum many-body systems whose coupling to an external bath is very weak. A particularly interesting class of such systems is those which do not thermalize under their own isolated quantum dynamics. In this review, we highlight the possibility for such systems to exhibit new non-equilibrium phases of matter. In particular, we focus on \"discrete time crystals\", which are many-body phases of matter characterized by a spontaneously broken discrete time translation symmetry. We give a definition of discrete time crystals from several points of view, emphasizing that they are a non-equilibrium phenomenon, which is stabilized by many-body interactions, with no analog in non-interacting systems. We explain the theory behind several proposed models of discrete time crystals, and compare a number of recent realizations, in different experimental contexts.\

Quantum computing leverages the quantum resources of superposition and entanglement to efficiently solve computational problems considered intractable for classical computers. Examples include calculating molecular and nuclear structure, simulating strongly-interacting electron systems, and modeling aspects of material function. While substantial theoretical advances have been made in mapping these problems to quantum algorithms, there remains a large gap between the resource requirements for solving such problems and the capabilities of currently available quantum hardware. Bridging this gap will require a co-design approach, where the expression of algorithms is developed in conjunction with the hardware itself to optimize execution. Here, we describe a scalable co-design framework for solving chemistry problems on a trapped ion quantum computer, and apply it to compute the ground-state energy of the water molecule. The robust operation of the trapped ion quantum computer yields energy estimates with errors approaching the chemical accuracy, which is the target threshold necessary for predicting the rates of chemical reaction dynamics.

We generalize past work on quantum sensor networks to show that, for d input parameters, entanglement can yield a factor O(d) improvement in mean squared error when estimating an analytic function of these parameters. We show that the protocol is optimal for qubit sensors, and conjecture an optimal protocol for photons passing through interferometers. Our protocol is also applicable to continuous variable measurements, such as one quadrature of a field operator. We outline a few potential applications, including calibration of laser operations in trapped ion quantum computing.

We examine the viability of quantum repeaters based on two-species trapped ion modules for long distance quantum key distribution. Repeater nodes comprised of ion-trap modules of co-trapped ions of distinct species are considered. The species used for communication qubits has excellent optical properties while the other longer lived species serves as a memory qubit in the modules. Each module interacts with the network only via single photons emitted by the communication ions. Coherent Coulomb interaction between ions is utilized to transfer quantum information between the communication and memory ions and to achieve entanglement swapping between two memory ions. We describe simple modular quantum repeater architectures realizable with the ion-trap modules and numerically study the dependence of the quantum key distribution rate on various experimental parameters, including coupling efficiency, gate infidelity, operation time and length of the elementary links. Our analysis suggests crucial improvements necessary in a physical implementation for co-trapped two-species ions to be a competitive platform in long-distance quantum communication.\

We demonstrate a Bayesian quantum game on an ion trap quantum computer with five qubits. The players share an entangled pair of qubits and perform rotations on their qubit as the strategy choice. Two five-qubit circuits are sufficient to run all 16 possible strategy choice sets in a game with four possible strategies. The data are then parsed into player types randomly in order to combine them classically into a Bayesian framework. We exhaustively compute the possible strategies of the game so that the experimental data can be used to solve for the Nash equilibria of the game directly. Then we compare the payoff at the Nash equilibria and location of phase-change-like transitions obtained from the experimental data to the theory, and study how it changes as a function of the amount of entanglement.

Trapped atomic ions are an ideal candidate for quantum network nodes, with long-lived identical qubit memories that can be locally entangled through their Coulomb interaction and remotely entangled through photonic channels. The integrity of this photonic interface is generally reliant on purity of single photons produced by the quantum memory. Here we demonstrate a single-photon source for quantum networking based on a trapped 138Ba+ ion with a single photon purity of g2(0)=(8.1\±2.3)\×10\−5 without background subtraction. We further optimize the tradeoff between the photonic generation rate and the memory-photon entanglement fidelity for the case of polarization photonic qubits by tailoring the spatial mode of the collected light.\

In an ion trap quantum computer, collective motional modes are used to entangle two or more qubits in order to execute multi-qubit logical gates. Any residual entanglement between the internal and motional states of the ions will result in decoherence errors, especially when there are many spectator ions in the crystal. We propose using a frequency-modulated (FM) driving force to minimize such errors and implement it experimentally. In simulation, we obtained an optimized FM gate that can suppress decoherence to less than 10\−4 and is robust against a frequency drift of more than \±1 kHz. The two-qubit gate was tested in a five-qubit trapped ion crystal, with 98.3(4)\% fidelity for a M{\o}lmer-S{\o}rensen entangling gate and 98.6(7)\% for a controlled-not (CNOT) gate. We also show an optimized FM two-qubit gate for 17 ions, proving the scalability of our method.

Quantum scrambling is the dispersal of local information into many-body quantum entanglements and correlations distributed throughout the entire system. This concept underlies the dynamics of thermalization in closed quantum systems, and more recently has emerged as a powerful tool for characterizing chaos in black holes. However, the direct experimental measurement of quantum scrambling is difficult, owing to the exponential complexity of ergodic many-body entangled states. One way to characterize quantum scrambling is to measure an out-of-time-ordered correlation function (OTOC); however, since scrambling leads to their decay, OTOCs do not generally discriminate between quantum scrambling and ordinary decoherence. Here, we implement a quantum circuit that provides a positive test for the scrambling features of a given unitary process. This approach conditionally teleports a quantum state through the circuit, providing an unambiguous litmus test for scrambling while projecting potential circuit errors into an ancillary observable. We engineer quantum scrambling processes through a tunable 3-qubit unitary operation as part of a 7-qubit circuit on an ion trap quantum computer. Measured teleportation fidelities are typically \∼80\%, and enable us to experimentally bound the scrambling-induced decay of the corresponding OTOC measurement.

Searching large databases is an important problem with broad applications. The Grover search algorithm provides a powerful method for quantum computers to perform searches with a quadratic speedup in the number of required database queries over classical computers. It is an optimal search algorithm for a quantum computer, and has further applications as a subroutine for other quantum algorithms. Searches with two qubits have been demonstrated on a variety of platforms and proposed for others, but larger search spaces have only been demonstrated on a non-scalable NMR system. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. The algorithm is performed for all 8 possible single-result oracles and all 28 possible two-result oracles. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state-marking scheme required to perform a classical search. All quantum solutions are shown to outperform their classical counterparts. We also report the first implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5\% and 89.6\%, respectively.

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device [1] with limited connectivity, and the other is a fully connected trapped-ion system [2]. Even though the two systems have different native quantum interactions, both can be programmed in a way that is blind to the underlying hardware, thus allowing the first comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that employ more connectivity clearly benefit from a better connected system of qubits. While the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that co-designing particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.

},
doi = {10.1073/pnas.1618020114},
url = {http://www.pnas.org/content/114/13/3305},
author = {N.M. Linke and Dmitri Maslov and Martin Roetteler and S. Debnath and C. Figgatt and K. A. Landsman and K. Wright and Christopher Monroe}
}
@article {1716,
title = {Co-Designing a Scalable Quantum Computer with Trapped Atomic Ions},
year = {2016},
month = {2016/02/09},
abstract = {The first generation of quantum computers are on the horizon, fabricated from quantum hardware platforms that may soon be able to tackle certain tasks that cannot be performed or modelled with conventional computers. These quantum devices will not likely be universal or fully programmable, but special-purpose processors whose hardware will be tightly co-designed with particular target applications. Trapped atomic ions are a leading platform for first generation quantum computers, but are also fundamentally scalable to more powerful general purpose devices in future generations. This is because trapped ion qubits are atomic clock standards that can be made identical to a part in 10^15, and their quantum circuit connectivity can be reconfigured through the use of external fields, without modifying the arrangement or architecture of the qubits themselves. In this article we show how a modular quantum computer of any size can be engineered from ion crystals, and how the wiring between ion trap qubits can be tailored to a variety of applications and quantum computing protocols.},
url = {http://arxiv.org/abs/1602.02840},
author = {Kenneth R. Brown and Jaewan Kim and Christopher Monroe}
}
@article {1695,
title = {Kaleidoscope of quantum phases in a long-range interacting spin-1 chain},
journal = {Physical Review B},
volume = {93},
year = {2016},
month = {2016/05/11},
pages = {205115},
abstract = {Motivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as 1/rα, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for α≲3, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U(1) continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future.
},
doi = {http://dx.doi.org/10.1103/PhysRevB.93.205115},
url = {http://arxiv.org/abs/1510.02108},
author = {Zhe-Xuan Gong and Mohammad F. Maghrebi and Anzi Hu and Michael Foss-Feig and Philip Richerme and Christopher Monroe and Alexey V. Gorshkov}
}
@article {1271,
title = {Many-body localization in a quantum simulator with programmable random disorder},
journal = {Nature Physics},
year = {2016},
month = {2016/06/06},
abstract = {

When a system thermalizes it loses all local memory of its initial conditions. This is a general feature of open systems and is well described by equilibrium statistical mechanics. Even within a closed (or reversible) quantum system, where unitary time evolution retains all information about its initial state, subsystems can still thermalize using the rest of the system as an effective heat bath. Exceptions to quantum thermalization have been predicted and observed, but typically require inherent symmetries or noninteracting particles in the presence of static disorder. The prediction of many-body localization (MBL), in which disordered quantum systems can fail to thermalize in spite of strong interactions and high excitation energy, was therefore surprising and has attracted considerable theoretical attention. Here we experimentally generate MBL states by applying an Ising Hamiltonian with long-range interactions and programmably random disorder to ten spins initialized far from equilibrium. We observe the essential signatures of MBL: memory retention of the initial state, a Poissonian distribution of energy level spacings, and entanglement growth in the system at long times. Our platform can be scaled to higher numbers of spins, where detailed modeling of MBL becomes impossible due to the complexity of representing such entangled quantum states. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems.

},
doi = {10.1038/nphys3783},
url = {http://arxiv.org/abs/1508.07026v1},
author = {Jacob Smith and Aaron Lee and Philip Richerme and Brian Neyenhuis and Paul W. Hess and Philipp Hauke and Markus Heyl and David A. Huse and Christopher Monroe}
}
@article {1202,
title = {Non-local propagation of correlations in long-range interacting quantum systems
},
journal = {Nature},
volume = {511},
year = {2014},
month = {2014/7/9},
pages = {198 - 201},
abstract = { The maximum speed with which information can propagate in a quantum many-body
system directly affects how quickly disparate parts of the system can become
correlated and how difficult the system will be to describe numerically. For
systems with only short-range interactions, Lieb and Robinson derived a
constant-velocity bound that limits correlations to within a linear effective
light cone. However, little is known about the propagation speed in systems
with long-range interactions, since the best long-range bound is too loose to
give the correct light-cone shape for any known spin model and since analytic
solutions rarely exist. In this work, we experimentally determine the spatial
and time-dependent correlations of a far-from-equilibrium quantum many-body
system evolving under a long-range Ising- or XY-model Hamiltonian. For several
different interaction ranges, we extract the shape of the light cone and
measure the velocity with which correlations propagate through the system. In
many cases we find increasing propagation velocities, which violate the
Lieb-Robinson prediction, and in one instance cannot be explained by any
existing theory. Our results demonstrate that even modestly-sized quantum
simulators are well-poised for studying complicated many-body systems that are
intractable to classical computation.
},
doi = {10.1038/nature13450},
url = {http://arxiv.org/abs/1401.5088v1},
author = {Philip Richerme and Zhe-Xuan Gong and Aaron Lee and Crystal Senko and Jacob Smith and Michael Foss-Feig and Spyridon Michalakis and Alexey V. Gorshkov and Christopher Monroe}
}
@article {1268,
title = {Experimental Performance of a Quantum Simulator: Optimizing Adiabatic Evolution and Identifying Many-Body Ground States
},
journal = {Physical Review A},
volume = {88},
year = {2013},
month = {2013/7/31},
abstract = { We use local adiabatic evolution to experimentally create and determine the
ground state spin ordering of a fully-connected Ising model with up to 14
spins. Local adiabatic evolution -- in which the system evolution rate is a
function of the instantaneous energy gap -- is found to maximize the ground
state probability compared with other adiabatic methods while only requiring
knowledge of the lowest $\sim N$ of the $2^N$ Hamiltonian eigenvalues. We also
demonstrate that the ground state ordering can be experimentally identified as
the most probable of all possible spin configurations, even when the evolution
is highly non-adiabatic.
},
doi = {10.1103/PhysRevA.88.012334},
url = {http://arxiv.org/abs/1305.2253v1},
author = {Philip Richerme and Crystal Senko and Jacob Smith and Aaron Lee and Simcha Korenblit and Christopher Monroe}
}
@article {1270,
title = {Quantum Catalysis of Magnetic Phase Transitions in a Quantum Simulator},
journal = {Physical Review Letters},
volume = {111},
year = {2013},
month = {2013/9/5},
abstract = { We control quantum fluctuations to create the ground state magnetic phases of
a classical Ising model with a tunable longitudinal magnetic field using a
system of 6 to 10 atomic ion spins. Due to the long-range Ising interactions,
the various ground state spin configurations are separated by multiple
first-order phase transitions, which in our zero temperature system cannot be
driven by thermal fluctuations. We instead use a transverse magnetic field as a
quantum catalyst to observe the first steps of the complete fractal devil{\textquoteright}s
staircase, which emerges in the thermodynamic limit and can be mapped to a
large number of many-body and energy-optimization problems.
},
doi = {10.1103/PhysRevLett.111.100506},
url = {http://arxiv.org/abs/1303.6983v2},
author = {Philip Richerme and Crystal Senko and Simcha Korenblit and Jacob Smith and Aaron Lee and Rajibul Islam and Wesley C. Campbell and Christopher Monroe}
}
@article {1492,
title = {Quantum Simulation of Spin Models on an Arbitrary Lattice with Trapped Ions
},
journal = {New Journal of Physics},
volume = {14},
year = {2012},
month = {2012/09/27},
pages = {095024},
abstract = { A collection of trapped atomic ions represents one of the most attractive
platforms for the quantum simulation of interacting spin networks and quantum
magnetism. Spin-dependent optical dipole forces applied to an ion crystal
create long-range effective spin-spin interactions and allow the simulation of
spin Hamiltonians that possess nontrivial phases and dynamics. Here we show how
appropriate design of laser fields can provide for arbitrary multidimensional
spin-spin interaction graphs even for the case of a linear spatial array of
ions. This scheme uses currently existing trap technology and is scalable to
levels where classical methods of simulation are intractable.
},
doi = {10.1088/1367-2630/14/9/095024},
url = {http://arxiv.org/abs/1201.0776v1},
author = {Simcha Korenblit and Dvir Kafri and Wess C. Campbell and Rajibul Islam and Emily E. Edwards and Zhe-Xuan Gong and Guin-Dar Lin and Luming Duan and Jungsang Kim and Kihwan Kim and Christopher Monroe}
}
@article {1269,
title = {Quantum Computing},
journal = {Nature},
volume = {464},
year = {2010},
month = {2010/3/4},
pages = {45 - 53},
abstract = { Quantum mechanics---the theory describing the fundamental workings of
nature---is famously counterintuitive: it predicts that a particle can be in
two places at the same time, and that two remote particles can be inextricably
and instantaneously linked. These predictions have been the topic of intense
metaphysical debate ever since the theory{\textquoteright}s inception early last century.
However, supreme predictive power combined with direct experimental observation
of some of these unusual phenomena leave little doubt as to its fundamental
correctness. In fact, without quantum mechanics we could not explain the
workings of a laser, nor indeed how a fridge magnet operates. Over the last
several decades quantum information science has emerged to seek answers to the
question: can we gain some advantage by storing, transmitting and processing
information encoded in systems that exhibit these unique quantum properties?
Today it is understood that the answer is yes. Many research groups around the
world are working towards one of the most ambitious goals humankind has ever
embarked upon: a quantum computer that promises to exponentially improve
computational power for particular tasks. A number of physical systems,
spanning much of modern physics, are being developed for this task---ranging
from single particles of light to superconducting circuits---and it is not yet
clear which, if any, will ultimately prove successful. Here we describe the
latest developments for each of the leading approaches and explain what the
major challenges are for the future.
},
doi = {10.1038/nature08812},
url = {http://arxiv.org/abs/1009.2267v1},
author = {Thaddeus D. Ladd and Fedor Jelezko and Raymond Laflamme and Yasunobu Nakamura and Christopher Monroe and Jeremy L. O{\textquoteright}Brien}
}
@article {1267,
title = {Protocol for Hybrid Entanglement Between a Trapped Atom and a Semiconductor Quantum Dot
},
journal = {Physical Review A},
volume = {80},
year = {2009},
month = {2009/12/30},
abstract = { We propose a quantum optical interface between an atomic and solid state
system. We show that quantum states in a single trapped atom can be entangled
with the states of a semiconductor quantum dot through their common interaction
with a classical laser field. The interference and detection of the resulting
scattered photons can then herald the entanglement of the disparate atomic and
solid-state quantum bits. We develop a protocol that can succeed despite a
significant mismatch in the radiative characteristics of the two matter-based
qubits. We study in detail a particular case of this interface applied to a
single trapped \Yb ion and a cavity-coupled InGaAs semiconductor quantum dot.
Entanglement fidelity and success rates are found to be robust to a broad range
of experimental nonideal effects such as dispersion mismatch, atom recoil, and
multi-photon scattering. We conclude that it should be possible to produce
highly entangled states of these complementary qubit systems under realistic
experimental conditions.
},
doi = {10.1103/PhysRevA.80.062330},
url = {http://arxiv.org/abs/0907.0444v1},
author = {Edo Waks and Christopher Monroe}
}